3 Answers
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That means it is a big volume of material (in this case cryogenic xenon) and subjected to a strong electric field. The field causes ionization electrons to drift to two or more non-colinear planes of detection wires. The front wires must be so-call "induction" wires that do not absorb the ionization electrons.

This means that you can get information about the drift electron's positions in at least two different direction and reconstruct their position in two dimension.

But it gets better: with a uniform field the drift velocity is very reliable, so if you know when the electrons started drifting and when they were detected you also know how far they drifted and thus have reconstructed their starting position in 3D.

Heavy noble gasses make a really good medium for such devices because

If sufficiently pure loose ionization electrons can go uncaptured for many miliseconds allowing very long drift distances (meters).

These materials scintillate (that is release light) when ionizing radiation passes through them. That light is detected within nanoseconds of it's release and used to tag the drift start time. The electron detection electronics, of course, tag the detection time.

LUX is searching for ionizing events in the detector that

Can not be explained by the (many) known kinds of physics that generate signals in these detectors

Have the characteristics that are expected of WIMP--ordinary matter interactions (which can be conjectured with some accuracy because we define a WIMP as having certain properties; basically there can only be elastic $Z^0$ at the WIMP vertex which generates a modest number of final states).

$\begingroup$Aside: I spent last summer helping to build a large microBooNE (a large liquid argon TPC for neutrino detection), so I have some sense of the difficulty of the project. Though LUX has much higher radio-purity requirements than microBooNE.$\endgroup$
– dmckee♦Nov 3 '13 at 17:14

First off, to say that WIMPs are the basis of dark matter is a little misleading. It's a well motivated model that can explain Dark Matter as well as be evidence of supersymmetry, but we have no experimental evidence to support it. It's still just a theory. Another well motivated candidate are Axions which help solve the Strong CP violation. ADMX is an experiment looking for these particles.

That being said, since we have to start somewhere, a WIMP is a great place to start because it solves two problems. So we have to understand how a WIMP interacts with regular nuclei (Xenon in LUX), and more importantly, how that interaction differs from most of the background. WIMP stands for weakly interacting massive particle which is a very descriptive name in addition to a great acronym. Weakly refers to the Weak nuclear force. When a WIMP comes in it will tend to interact with the nuclei, while most backgrounds interact with the electrons.

The interactions give off energy in two ways: Scintillation and Ionization. The prompt scintillation light (S1) is measured immediately, but the electrons drift through the liquid until they reach the top high voltage electrode where they scintillate (S2). The ratio of energy (S1/S2) is different in the two pathways, so LUX is able to distinguish between Nuclear Recoils (WIMPs and relatively rare backgrounds) and Electron Recoils (the majority of standard backgrounds). This discrimination is one of the essential tools for finding the still hypothetical WIMP.

This question is a really interesting one.
Firstly I feel the need to point out that Weakly Interacting Massive Particles, which you claim are considered to be a basis of Dark Matter, MAY NOT BE a basis of dark matter(purely my thought) as the assumption is based on the knowledge of regular matter, which consists of roughly 4 percent of the universe.

Now assuming that WIMPs do indeed serve as a basis of Dark Matter, we know that WIMPs are theorised as weakly interacting, that is to say they fly around the universe without hitting anything, but there might be one unlucky WIMP who hits head-on with a Xenon nucleus in the LUX detector, a flash of light will be emitted due to it and you may well shout,"Eureka, we have dark matter in company!"